Explanation Given info: The data represents the results of the regression analysis based on a sample of 15 observations of the variables filter application temperature ( x ) and % efficiency of BOD removal ( y ) . The summary statistics are as follows: ∑ x i = 402 , ∑ x i 2 = 11 , 098 , s = 3.725 and β ^ 1 = 1.7035 . At 0.01 level of significance the test hypothesis is H 0 : β 1 = 1 versus H 1 : β 1 > 1 . Furthermore, it is given that β ′ 1 = 2 and σ = 4 . Calculation: The test hypotheses are given below: Null hypothesis: H 0 : β 1 = 1 That is, the expected change in % efficiency of BOD removal associated with 1% increase in filter application temperature is 1. Alternative hypothesis: H 1 : β 1 > 1 That is, the expected change in % efficiency of BOD removal associated with 1% increase in filter application temperature is not equal to 1. T-test statistic: The test statistic is, t = β ^ 1 − β 1 s β ^ 1 ∼ t ( n − 2 ) Degrees of freedom: The sample size is n = 15 The degrees of freedom is, d . f = n − 2 = 15 − 2 = 13 Thus, the degree of freedom is 13. Level of significance: The level of significance is α = 0.01 . Critical value : Software procedure: Step by step procedure to obtain the critical value using the MINITAB software: Choose Graph > Probability Distribution Plot choose View Probability > OK. From Distribution , choose ‘ t ’ distribution and enter 13 as degrees of freedom . Click the Shaded Area tab. Choose Probability Value and Right Tail for the region of the curve to shade. Enter the Probability value as 0.01 . Click OK . Output using the MINITAB software is given below: From the output, the critical value is 2.650. Thus, the critical value is ( t 0.01 , 13 ) = 2.650 . Estimate of error standard deviation of slope coefficient: The general formula for the estimate of error standard deviation of slope coefficient is, σ β ^ 1 = σ S x x , The defining formula for S xx is, S x x = ∑ i x i 2 − ( ∑ i x i ) 2 n The estimate of error standard deviation of slope coefficient is, s β ^ 1 = s S x x = s ∑ i x i 2 − ( ∑ i x i ) 2 n = 3

Probability and Statistics for Engineering and the Sciences

9th Edition

ISBN: 9781305251809

Author: Jay L. Devore

Publisher: Cengage Learning

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Chapter 12.3, Problem 43E

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Obtain the probability of type II error β in the given situation.

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Probability and Statistics for Engineering and the Sciences

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Solution Summary: The author compares the probability of type II error beta and the expected change in % efficiency of BOD removal at 0.01 level of significance.

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Chapter 12 Solutions